Variational Quantum Computation of Excited States

Oscar Higgott1,2, Daochen Wang1,3, and Stephen Brierley1

1Riverlane, 3 Charles Babbage Road, Cambridge CB3 0GT
2Department of Physics and Astronomy, University College London, London, WC1E 6BT
3Joint Center for Quantum Information and Computer Science, University of Maryland, College Park, MD 20742

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Abstract

The calculation of excited state energies of electronic structure Hamiltonians has many important applications, such as the calculation of optical spectra and reaction rates. While low-depth quantum algorithms, such as the variational quantum eigenvalue solver (VQE), have been used to determine ground state energies, methods for calculating excited states currently involve the implementation of high-depth controlled-unitaries or a large number of additional samples. Here we show how overlap estimation can be used to deflate eigenstates once they are found, enabling the calculation of excited state energies and their degeneracies. We propose an implementation that requires the same number of qubits as VQE and at most twice the circuit depth. Our method is robust to control errors, is compatible with error-mitigation strategies and can be implemented on near-term quantum computers.

Small quantum computers will soon be available with the capability of performing some computational tasks that are beyond the reach of even the largest classical supercomputers. However, these devices will be noisy, limiting the complexity of the algorithms that they can implement correctly. We present a new quantum algorithm that can calculate the spectrum of complex quantum systems, and show how it may be used to perform useful computations even on the imperfect quantum devices available in the near- future.
The algorithm we have developed - variational quantum deflation - uses a hybrid of both quantum and classical resources to determine the excited state energies of quantum systems. We achieve this at almost no extra cost over the most promising hybrid method for determining ground state energies - the variational quantum eigensolver - by first finding the ground state and then energetically exciting it, allowing the first excited state to be found as the ground state of the new modified system. By repeating this procedure and incorporating techniques to mitigate device errors, our method can determine multiple excited states even on imperfect hardware. Our algorithm is therefore a promising candidate for useful near-term quantum-enhanced computation.

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[63] Utkarsh Azad and Animesh Sinha, "qLEET: visualizing loss landscapes, expressibility, entangling power and training trajectories for parameterized quantum circuits", Quantum Information Processing 22 6, 256 (2023).

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[68] Shigeki Gocho, Hajime Nakamura, Shu Kanno, Qi Gao, Takao Kobayashi, Taichi Inagaki, and Miho Hatanaka, "Excited state calculations using variational quantum eigensolver with spin-restricted ansätze and automatically-adjusted constraints", npj Computational Materials 9 1, 13 (2023).

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[70] Andy C. Y. Li, M. Sohaib Alam, Thomas Iadecola, Ammar Jahin, Joshua Job, Doga Murat Kurkcuoglu, Richard Li, Peter P. Orth, A. Barış Özgüler, Gabriel N. Perdue, and Norm M. Tubman, "Benchmarking variational quantum eigensolvers for the square-octagon-lattice Kitaev model", Physical Review Research 5 3, 033071 (2023).

[71] Shi-Xin Zhang, Chang-Yu Hsieh, Shengyu Zhang, and Hong Yao, "Neural predictor based quantum architecture search", Machine Learning: Science and Technology 2 4, 045027 (2021).

[72] Chufan Lyu, Xiaoyu Tang, Junning Li, Xusheng Xu, Man-Hong Yung, and Abolfazl Bayat, "Variational quantum simulation of long-range interacting systems", New Journal of Physics 25 5, 053022 (2023).

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[81] Alexey Pyrkov, Alex Aliper, Dmitry Bezrukov, Yen-Chu Lin, Daniil Polykovskiy, Petrina Kamya, Feng Ren, and Alex Zhavoronkov, "Quantum computing for near-term applications in generative chemistry and drug discovery", Drug Discovery Today 28 8, 103675 (2023).

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[83] Yu Zhang, Lukasz Cincio, Christian F. A. Negre, Piotr Czarnik, Patrick J. Coles, Petr M. Anisimov, Susan M. Mniszewski, Sergei Tretiak, and Pavel A. Dub, "Variational quantum eigensolver with reduced circuit complexity", npj Quantum Information 8 1, 96 (2022).

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[95] Daniel Huerga, "Variational Quantum Simulation of Valence-Bond Solids", Quantum 6, 874 (2022).

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[111] Sandip Maiti, Debasish Banerjee, Bipasha Chakraborty, and Emilie Huffman, "Spontaneous symmetry breaking in a SO(3) non-Abelian lattice gauge theory in 2+1D with quantum algorithms", Physical Review Research 7 1, 013283 (2025).

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[113] Zhijie Sun, Xiaopeng Li, Jie Liu, Zhenyu Li, and Jinlong Yang, "Circuit-Efficient Qubit Excitation-Based Variational Quantum Eigensolver", Journal of Chemical Theory and Computation acs.jctc.5c00119 (2025).

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[117] Maria-Andreea Filip, "Fighting Noise with Noise: A Stochastic Projective Quantum Eigensolver", Journal of Chemical Theory and Computation 20 14, 5964 (2024).

[118] Giuseppe Clemente, Arianna Crippa, Karl Jansen, and Cenk Tüysüz, Quantum Computer Music 433 (2022) ISBN:978-3-031-13908-6.

[119] Adrián Pérez-Salinas, Juan Cruz-Martinez, Abdulla A. Alhajri, and Stefano Carrazza, "Determining the proton content with a quantum computer", Physical Review D 103 3, 034027 (2021).

[120] Nahum Sá, Ivan S. Oliveira, and Itzhak Roditi, "Towards solving the BCS Hamiltonian gap in near-term quantum computers", Results in Physics 44, 106131 (2023).

[121] Kübra Yeter-Aydeniz, Raphael C. Pooser, and George Siopsis, "Practical quantum computation of chemical and nuclear energy levels using quantum imaginary time evolution and Lanczos algorithms", npj Quantum Information 6 1, 63 (2020).

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[124] Dan-Bo Zhang, Hongxi Xing, Hui Yan, Enke Wang, and Shi-Liang Zhu, "Selected topics of quantum computing for nuclear physics* ", Chinese Physics B 30 2, 020306 (2021).

[125] Anthony W. Schlimgen, Kade Head-Marsden, LeeAnn M. Sager, Prineha Narang, and David A. Mazziotti, "Quantum simulation of the Lindblad equation using a unitary decomposition of operators", Physical Review Research 4 2, 023216 (2022).

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[129] Joel Bierman, Yingzhou Li, and Jianfeng Lu, "Quantum Orbital Minimization Method for Excited States Calculation on a Quantum Computer", Journal of Chemical Theory and Computation 18 8, 4674 (2022).

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[152] Martin R. Albrecht, Miloš Prokop, Yixin Shen, and Petros Wallden, "Variational quantum solutions to the Shortest Vector Problem", Quantum 7, 933 (2023).

[153] Qing Guo and Ping-Xing Chen, "Optimization of VQE-UCC Algorithm Based on Spin State Symmetry", Frontiers in Physics 9, 735321 (2021).

[154] Annie E. Paine, Vincent E. Elfving, and Oleksandr Kyriienko, "Quantum kernel methods for solving regression problems and differential equations", Physical Review A 107 3, 032428 (2023).

[155] Chris N. Self, Kiran E. Khosla, Alistair W. R. Smith, Frédéric Sauvage, Peter D. Haynes, Johannes Knolle, Florian Mintert, and M. S. Kim, "Variational quantum algorithm with information sharing", npj Quantum Information 7 1, 116 (2021).

[156] Qi Gao, Gavin O. Jones, Mario Motta, Michihiko Sugawara, Hiroshi C. Watanabe, Takao Kobayashi, Eriko Watanabe, Yu-ya Ohnishi, Hajime Nakamura, and Naoki Yamamoto, "Applications of quantum computing for investigations of electronic transitions in phenylsulfonyl-carbazole TADF emitters", npj Computational Materials 7 1, 70 (2021).

[157] Zhimin He, Xuefen Zhang, Chuangtao Chen, Zhiming Huang, Yan Zhou, and Haozhen Situ, "A GNN-based predictor for quantum architecture search", Quantum Information Processing 22 2, 128 (2023).

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[159] Kaoru Mizuta, Mikiya Fujii, Shigeki Fujii, Kazuhide Ichikawa, Yutaka Imamura, Yukihiro Okuno, and Yuya O. Nakagawa, "Deep variational quantum eigensolver for excited states and its application to quantum chemistry calculation of periodic materials", Physical Review Research 3 4, 043121 (2021).

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[162] Philipp Schleich, Joseph Boen, Lukasz Cincio, Abhinav Anand, Jakob S. Kottmann, Sergei Tretiak, Pavel A. Dub, and Alán Aspuru-Guzik, "Partitioning Quantum Chemistry Simulations with Clifford Circuits", Journal of Chemical Theory and Computation 19 15, 4952 (2023).

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[174] Saad Yalouz, Emiel Koridon, Bruno Senjean, Benjamin Lasorne, Francesco Buda, and Lucas Visscher, "Analytical Nonadiabatic Couplings and Gradients within the State-Averaged Orbital-Optimized Variational Quantum Eigensolver", Journal of Chemical Theory and Computation 18 2, 776 (2022).

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